TameFusion

The material here is taken from the paper "Tame Fusion" by S.D. Scott (Algebra Colloquium 10 (2003) 543--566).

Scott discussed certain types of fusion in tame nearrings in this paper.

Let $N$ be a (left) zero-symmetric nearring with an identity.

A right $N$-subgroup $M$ of $N$ is self monogenic if there exists an element $\alpha\in M$ such that $\alpha M=M$.
A (unital) $N$-group $V$ is said to have property $q$ if either $V$ has no maximal $N$-subgroups or all maximal $N$-subgroups are submodules.
For a set $S$ of minimal ideals of $N$, define $mr(S)$ to be the set of all right $N$ subgroups $M$ of $N$ such that $M$ is minimal with respective to the property that $MB\not=0$ for all $B\in S$. In case that $S=\varnothing$, $mr(S)=\varnothing$. If $S=\{A\}$, where $A$ is a minimal ideal of $N$, we write $mr(A)$ for $mr(\{A\})$.

It is shown that

Let $N$ be a compatible nearring $N$ with descending chain condition on right ideals (DCCR). Then $N$ is $N$-nilpotent as an $N$-group if and only if $N$ has property $q$.
If $N$ has descending chain condition on $N$-subgroups (DCCN) and $A$ is a minimal ideal of $N$, then all elements of $mr(A)$ are self monogenic and $N$-isomorphic. Furthermore, a right $N$-subgroup of $N$ which is $N$-isomorphic to an element of $mr(A)$ is in $mr(A)$.
If $N$ is a tame nearring with DCCR and $S$ a nonempty collection of minimal ideals of $N$, then $mr(S)$ is nonempty and all elements of $mr(S)$ has property $q$. Moreover
- $M\in mr(S)$ implies that for all $B\in S$, there is an $M_B\in mr(B)$ with $M_B\subseteq M$.
- $M=\langle M_B\mid B\in S\rangle$. Thus, $M$ may be regarded as a "fusion product" of $M_B$, $B\in S$.

The following types of fusion are defined and studied in tame nearrings.

We saying that fusion is self monogenic in $N$ if for any given set $S$ of minimal ideals of $N$, every element of $mr(S)$ is self monogenic. Similarly, we say fusion is exact (in $N$) if all elements of $mr(S)$ are $N$-isomorphic. Finally, we call fusion minimal if the elements of $mr(S)$ are generated by minimal non-nilpotent right $N$-subgroups.

Main Theorem In a tame nearring $N$ with DCCR, (1) fusion is self monogenic, (2) fusion is exact, and (3) fusion is minimal.

MR0648900 (83f:16053)

Scott, S. D.

Zero sets---consequences for primitive near-rings.

Proc. Edinburgh Math. Soc. (2) 25 (1982), no. 1, 55--63.

Let $N$ be a left near-ring (not necessarily zero-symmetric) and let $V$ be an $N$-group. If $S\subseteq N$ then $Z(S)\colon=\{v\in V\colon vs=0$ for all $s\in S\}$ is called the zero subset of $V$ with respect to $S$. Since $Z(S_1)=Z(S_2)$ if and only if these zero sets have equal annihilators, zero sets correspond to annihilators (in a Galois-type way). If $N$ is 2-primitive and $N_0$ not a ring then the zero sets provide a "$Z$-topology" for $V$ such that all maps $v\rightarrow vn$ are continuous. This topology and a related other "$Z^t$-topology" (obtained by adding all constants to $N$) are used to study compatible $N$-groups $V$ (i.e., for all $v\in V$ and $n\in N$ there is some $m\in N$ with $(v+w)n-vn=wm$ for all $w\in V$) and compatible near-rings (i.e., those having faithful, compatible, unitary $N$-groups). For instance, it is shown that a 2-primitive compatible nonring $N$ is discrete if and only if $N$ has a minimal right ideal. If the nonring $N$ is primitive and compatible on $V$ with a.c.c. on right ideals then either $V$ is finite and $N\cong M_0(V)$ or the $Z^t$-topology is sparse. Finally, these results are applied to (compatible) near-rings of polynomial mappings on simple $\Omega$-groups $V$. If $V$ is a field, the $Z^t$-topology is just the Zariski topology.

Reviewed by G?Pilz

MR0620925 (83b:16032)

Scott, S. D.

Tame near-rings and $N$-groups.

Proc. Edinburgh Math. Soc. (2) 23 (1980), no. 3, 275--296.

Throughout this review any near-ring $N$ will be left distributive, zero symmetric and have an identity. Furthermore, all $N$-groups will be unitary. An $N$-group $V$ is said to be tame if every $N$-subgroup of $V$ is a submodule. The near-ring $N$ itself is said to be tame if $N$ possesses a faithful tame $N$-group. Tame near-rings and $N$-groups were first investigated by the author ["Near-rings and near-ring modules", Ph.D. Thesis, Australian National Univ., Canberra, 1970]. Many of the results of this paper may be found in this dissertation, although the proofs given here are often simpler. From the many interesting results of this paper we mention the following: (1) The sum of all nilpotent right ideals of a tame near-ring $N$ is an ideal of $N$. (2) Let $N$ be a near-ring with a faithful tame $N$-group $V$. If $N=A\oplus B$, where $A$ and $B$ are ideals of $N$, then $V=U\oplus W$ with $U$ and $W$ submodules of $V$ such that $(0\colon W)=A$ and $(0\colon U)=B$. (3) Let $N$ be a near-ring with maximal condition on right ideals and $V$ a tame $N$-group. If $V$ is finitely generated, then any submodule of $V$ is finitely generated. (4) If $N$ is a tame near-ring with minimal condition on right ideals, then the radical $J_2(N)$ is nilpotent. (5) If $N$ is a tame near-ring with minimal condition on right ideals, then $N$ satisfies even the minimal and maximal condition on right $N$-subgroups.

The author then proceeds to define compatible and 2-tame $N$-groups and near-rings. Let again $N$ be a near-ring and $V$ be an $N$-group. $V$ is said to be 2-tame if for each $v\in V$ and $\alpha\in N$ there exists $\beta\in N$ such that $(v+w_i)\alpha-v\alpha=w_i\beta$ for any two elements $w_i$, $i=1,2$, of $V$. And $N$ is called a 2-tame near-ring if $N$ has a faithful 2-tame $N$-group. (6) If $N$ is a near-ring with a 2-tame $N$-group $V$, where $V$ has maximal and minimal condition on submodules, then a Krull-Shmidt theorem holds on $V$ (uniqueness of a decomposition of $V$ as finite direct sum of indecomposables). (7) If $N$ is a 2-tame near-ring with minimal condition on right ideals, and if no nonzero homomorphic image of $N$ is a ring, then $N$ is finite. Finally, the author studies "centralisers of abelian submodules". The results in this context are used to prove: (8) If the near-ring $I(V)$ generated by all inner automorphisms of a group $V$ satisfies the minimal condition on right ideals, then $I(V)$ is finite.

Reviewed by G. Betsch